A Polycyclic Presentation for the q-Tensor Square of a Polycyclic Group

Let $G$ be a group and $q$ a non-negative integer. We denote by ${\nu }^q(G)$ a certain extension of the $q$-tensor square $G{\otimes }^{q}G$ by $G\times G$. In this paper we derive a polycyclic presentation for $G{\otimes }^{q}G$, when $G$ is polycyclic, via its embedding into ${\nu }^q(G)$. Furthermore, we derive presentations for the $q$-exterior square $G{\wedge }^{q}G$ and for the second homology group $H_2(G,{\mathbb{Z}}_q)\mathrm{.}$ Additionally, we establish a criterion for computing the $q-$exterior centre $Z_q^{\wedge }(G)$ of a polycyclic group $G,$ which is helpful for deciding whether $G$ is capable modulo $q$. These results extend to all $q\ge 0$ existing methods due to Eick and Nickel for the case $q=0$.